All Questions
Tagged with pr.probability st.statistics
1,134 questions
3
votes
1
answer
467
views
How to perform Importance Sampling with Prior Information
Let us define a random variable $X$ with density function $p(x)$. We wish to calculate $\mathbb{E}[f(X)] = \int f(x)p(x)dx$. We can compute the expectation by Monte Carlo simulations as
$$\mathbb{E}[...
- 3,979
18
votes
1
answer
2k
views
Gini Coefficient and Renyi Entropy
Gini coefficient (aka Gini Index) is a quantity used in economics to describe income inequality. It is 0 for uniformly distributed income, and approaches 1 when all income is in hands of one ...
CommunityBot
- 1
1
vote
1
answer
2k
views
Sum of covariance matrix of products of dependent variables
Consider the sequences of random variables $\{X_i\}_{i=1}^n$ and $\{Y_i\}_{i=1}^n$, as well as the corresponding sequence of products, $\{X_i Y_i\}_{i=1}^n$. All $X_i$ share the same mean value, $\...
- 965
4
votes
2
answers
327
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Estimate on gaussian distribution
Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that
$$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$
and such that $...
- 293
2
votes
1
answer
591
views
Concentration rates for the posterior distribution
Sanov's theorem and Dvoretzky–Kiefer–Wolfowitz's inequality tell us how fast the empirical distribution concentrates around the true underlying probabilty distribution.
What is known about the ...
CommunityBot
- 1
3
votes
0
answers
494
views
Maximization of a total variation distance subject to another total variation distance in Markov chain
Suppose two dependent random variables $X$ and $V$ from finite alphabets $\mathcal{V}$ and $\mathcal{X}$ with known joint and marginal distributions are given. Let $P_{XV}$ and $P_X$ and $P_V$ are the ...
- 1,109
6
votes
1
answer
375
views
Deviation bound for the maximum of the norm of Wiener process
Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof:
$$
{...
- 29.8k
4
votes
1
answer
189
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Weak ergodicity of nonhomogenous products of 0-1 matrices
Here is a question which probably has a negative answer, but I couldn't find any literature directly on it.
Let $(A_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted ...
- 23.2k
0
votes
1
answer
221
views
Expected rank of players in a Bradley-Terry round-robin tournament
Let $[n]$=$\{1,\dots,n\}$ be a set of players in a round-robin tournament. Each player $i$ has an associated skill parameter, $\lambda_{i}$, and the probability that player $i$ defeats player $j$ is $\...
- 28k
3
votes
1
answer
528
views
Cover a line segment randomly with smaller line segments
Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon).
But the problem when the circle is changed to a line segment doesn't seem to have been ...
- 161
1
vote
0
answers
101
views
What is the range of a positive random variable after whitening?
Let ${\bf x}\in\mathbb R^N$ be a positive multivariate random variable, i.e.
$$x_i\in [0,\infty).$$
What is the range after whitening, i.e. the range of ${\bf y} = \sqrt{C}^{-1}{\bf x}$ with the ...
- 24.8k
4
votes
2
answers
462
views
Bounding the tail of an average using the the tail of individual members
Let $X_1,X_2,\ldots,X_n$ be an i.i.d. sequence of $n$ positive random variables with mean $E[X_1]=\mu_X<\infty$ and the second moment $E[X_1^2]=\infty$.
I am interested in upper-bounding $P\left(...
CommunityBot
- 1
1
vote
0
answers
132
views
Eigen value distribution of autocorrelated Wishart matrix
Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...
- 11
4
votes
2
answers
5k
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Proof of Von Neumann's debiasing algorithm
Assume you have a source of random binary information that has a bias but no correlation between consecutive bits. John von Neumann describes an algorithm to debias the random source and output a ...
- 1,676
4
votes
1
answer
288
views
Equivalent method for maximum likelihood estimation of covariance parameters
My goal is to estimate the parameters of a covariance matrix $\Omega$, by maximizing the following log-likelihood function:
$$\log L(\vec\tau, \rho, \sigma \mid W, X) = -m\ln(\left | \Omega \right |) ...
7
votes
2
answers
2k
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What is the maximum entropy distribution on the natural numbers?
On the reals $\mathbb{R}$, the maximum entropy distribution with a given mean and variance is the Gaussian distribution.
Let $\mu, \sigma > 0$. What is the maximum entropy distribution on the ...
- 8,501
1
vote
0
answers
98
views
Small ball probabilities for functions of correlated normals
Let $f : \mathbb{R}^k \rightarrow \mathbb{R}$ and let $X$ be distributed k-dimensional normal with mean $0$ (with "arbitrary" covariance matrix). I am looking for references with bounds of the form: ...
- 111
4
votes
0
answers
988
views
Probability distribution function for singular value sum of Gaussian random matrix
Let $\mathbf{X}$ be an $N \times N$ random matrix with IID Gaussian entries. They can be standard normal, but $N$ is not large: that is $N$ $<$ 6, typically. Call its singular value decomposition (...
2
votes
1
answer
356
views
The first eigenvalue of a branching process matrix
Let $M$ be the real square matrix of a typed branching process, such that $M_{ij}$ is the expected value of offspring of type $j$ emanating from type $i$.
We know that if the first eigenvalue if $M$ ...
CommunityBot
- 1
4
votes
1
answer
234
views
Statistical models in terms of families of random variables
A statistical model is a function $P : \Theta \to \Delta(X)$, where $\Theta$ is a parameter space, and $\Delta(X)$ is the set of probability measures on a state space $X$.
Suppose that $\Theta$ and $...
- 1,989
-2
votes
1
answer
347
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Forms of multivariate CLT [closed]
I am looking for a good reference for differnt kinds of multivariate central limit theorems. I was wondering how far the i.i.d. condition of the standard multivariate clt can be relaxed, as in can the ...
- 7,514
6
votes
0
answers
189
views
Pettis Integrability and Laws of Large Numbers
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
- 3,958
0
votes
0
answers
213
views
Behavior of the sum of the exponents of chi-squared random variables normalized by their maximum
Let $X_1,X_2,\ldots,X_n$ be a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom, and denote by $X_\max$ the maximum of this sequence. Furthermore, let $k=\omega(1)$ ...
CommunityBot
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3
votes
1
answer
188
views
Markov Chains based on sampled transition probabilities [closed]
If I have a process that transitions between states with some set, unknown probability, I can sample to find the transition probability. This probability is a sample average, with a well understood ...
- 1,902
2
votes
1
answer
557
views
Is this a closed set?
Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of ...
- 60.5k
4
votes
1
answer
193
views
Variance of central limit distribution for $P(x) \sim 1/x^{1+\alpha}$ for finite but large $N$?
Is it known what the next-to-leading order term is in the variance of the central limit distribution for the average of $N$ variables each of which is distributed according to $P(x) \sim 1/x^{1+\alpha}...
- 188k
4
votes
1
answer
704
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Central limit theorem for $P(x)\sim 1/x^3$ distribution
I have a random variable $x \in (0,\infty)$ with distribution $P(x)$ falling off slowly $P(x) \sim 1/x^3$ for large $x$. So the expectation value $\bar{x}$ is finite but the second moment $\bar{x^2}$ ...
- 1,150
10
votes
2
answers
590
views
"Fractional sampling" from a probability distribution
My question concerns an operation on probability distributions which has arisen in some applied research. It is well-defined mathematically (at least in a limited context), but I don't know how to ...
- 7,514
1
vote
0
answers
100
views
Distribute Monte Carlo samples among dimensions
Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...
- 101
2
votes
1
answer
871
views
Maximum of a sequence of $n$ positive random variables where variance is an increasing function of $n$
Suppose I have a sequence of $n$ i.i.d. random variables $X_1,X_2,\ldots,X_n$. Each $X_i$ is positive and has variance $\sigma(n)$ that is an increasing function of the number of variables in the ...
- 456
3
votes
1
answer
651
views
What conditions on a probability distribution defined by long-time averaging do I need to satisfy a central limit theorem?
For integer $n$, $1 \le n \le N$, consider the random variables
$X_n = \cos[t \omega_n]$
For any fixed $N$, we can take the mean
$Y_N = \frac{1}{N} \sum_{n=1}^N X_n$
and define a (cumulative) ...
- 19.3k
4
votes
1
answer
5k
views
Asymptotic behavior of max of chi-squared distribution
Suppose $X_{\max}$ is the maximum in a sequence $X_1,X_2,\ldots,X_n$ where each $X_i\sim\chi^2_k$ is an i.i.d. chi-squared random variable with $k$ degrees of freedom.
Since chi squared distribution ...
- 7,514
1
vote
0
answers
463
views
How far away is the maximum of $n$ i.i.d. chi-squared random variables from the rest of the sequence as $n$ gets large?
Suppose that I have a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom $X_1, X_2, \ldots, X_n$, and denote $X_{\max}=\max(X_1, X_2, \ldots, X_n)$. Let $k$ be increasing ...
- 907
0
votes
1
answer
369
views
How to calculate eigenvalue density function of $XX^\dagger$ from the density function of X
Let X be a complex random matrix, which has the probability function (drawn from the ensemble) V($XX^\dagger$), where V(x) is some function which guaranties good behavior at infinity. Note the unitary ...
- 7,514
13
votes
1
answer
3k
views
What is the maximum-entropy distribution given mean, variance, skewness, and kurtosis?
$X\in \mathbb{R}$. Which distribution $P(X)$ has the highest possible entropy given its expected value, variance, skewness, and kurtosis? Is it an exponential family distribution of the form $P(X) \...
CommunityBot
- 1
6
votes
2
answers
2k
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Absolute moments of symmetrical distributions
Suppose $F~$ is a probability distribution symmetrical about 0, for which all moments exist. Let $\mu_i~$be the $i$-th moment (of course $\mu_i=0$ if $i~$ is odd).
We know there are some conditions ...
- 60.5k
0
votes
1
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514
views
Relating percentiles to moments [closed]
There are at least two ways people look at statistical data:
A. For mathematicians, scientists, engineers, economists and such the most familiar distribution parameters would be analytical: mean, ...
- 91.8k
4
votes
1
answer
159
views
diffusions corresponding to estimators
I am an undergraduate math student preparing my thesis. Currently I am reading L.D Brown's (1971) paper Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems. Here is a ...
CommunityBot
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2
votes
1
answer
3k
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Empirical estimator fot the total variation distance on a finite space
I have two probability measures $p$ and $p'$ on a finite set $X$ which I do not know precisely, but which I can sample from. I would like to estimate their total variation (omitting multiplier $2$):
$$...
- 7,514
1
vote
0
answers
186
views
Shrinkage (or Stein's phenomenon) in low dimensions, discrete contexts
I am trying to understand shrinkage, or the Stein phenomenon. As someone without a statistics background, the focus in most introductory presentations on normal distributions and squared error loss ...
- 203
7
votes
2
answers
649
views
What's the standard name for sets of a given size with maximal probability (or a given probability and minimal size)?
The definition I'm going to give isn't quite the concept I really want, but it's a good approximation. I don't want to make the definition too technical and specific because if there's a standard name ...
- 833
2
votes
0
answers
341
views
Marginalizing multivariate normal over defined interval
Hello everyone,
I am trying to obtain an analytic expression for the following Gaussian integral
$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} \;...
- 101
3
votes
1
answer
1k
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Probability Density Optimization
I am working on an optimization problem which I am stuck on towards the end.
Essentially, I have two probability density functions in $\mathbb{R}^2$, call them $q(x,y)$ and $p(x,y)$, now I define ...
CommunityBot
- 1
0
votes
2
answers
429
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E[log(Z_t^2)], proof of convergence with Law of Large Numbers
Hi all,
question:
Let $Z_t$ be an iid sequence with $$\mathbb{E}\log(Z_t^2)<0 $$
Show that $$\sum_{j=0}^\infty Z_t^2 Z_{t-1}^2 ... Z_{t-j}^2 < \infty$$ almost surely
I am supposed to use LLN ...
- 5,721
4
votes
0
answers
980
views
Inverse Fourier Transform involving a Bessel Function, Exponential, and Power
I'm interested in this integral as a function of $r$ for various spectral densities $S(s)$:
$\frac{2 \pi}{r^{p/2}-1} \int_{0}^{\infty} S(s) J_{p/2-1}(2 \pi r s) s^{p/2} ds $, where $J_{p/2-1}$ is a ...
- 41
4
votes
3
answers
3k
views
What is the name for a non-normalized distribution?
For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work ...
CommunityBot
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1
vote
1
answer
16k
views
Calculating $E[X^2Y^2]$ given $E[X^2]$, $E[Y^2]$, $E[X]$, $E[Y]$, and that $X$, $Y$ are Gaussian. [closed]
Suppose $E[X]=E[Y]=0$, and $E[X^2]=E[Y^2]=1$. Can you show that $E[X^2Y^2] = 1 + 2\operatorname{cov}(X,Y)^2$? I am not even sure if this expression is correct, I found it in a geostatistics paper, ...
- 3,958
-1
votes
1
answer
1k
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Rank of covariance matrix whose diagonal elements are same [closed]
Suppose A is a covariance matrix whose diagonal elements are same, i.e. $A_{1,1}=A_{2,2}=\cdots=A_{N,N}$, can we conclude that A is full rank?
Suppose the absolute values of the off-diagonal elements ...
- 2,361
1
vote
1
answer
405
views
Convergence to a k-dimensional Gaussian vector
Suppose I have a sequence of stochastic processes $X_{N}(t)$, $N=1,2,3,\ldots$ with mean zero and that I know for every fixed $t$, the random variable $X_{N}(t)$ converges in law to a Gaussian random ...
- 148k
10
votes
3
answers
1k
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Rapid evaluation of multivariate normal integral
I'm implementing a model that requires me to numerically evaluate a multivariate normal integral of the following form
$$\int_{-\infty}^\infty \phi(z)\displaystyle\prod_{i=1}^N \Phi(a_iz+b_i) \, dz,$...
- 101